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Chapter 3: Problem 3

Without using a calculator or computer, determine which of the two numbers \(2^{125}\) and \(32 \cdot 10^{36}\) is larger.

Short Answer

Expert verified
The number \(32 \cdot 10^{36}\) is larger than \(2^{125}\), because after simplifying both expressions, we have \(2^{41} \cdot 5^{36}\) which is greater than \(2^{125}\) due to the presence of the \(5^{36}\) term.

Step by step solution

01

Simplify \(32 \cdot 10^{36}\)

We can write \(32\) as a power of \(2\), since \(2^5=32\). Then the expression becomes: \[2^5 \cdot 10^{36}\]
02

Notice that both numbers have a base of 2

We can rewrite \(2^5 \cdot 10^{36}\) to have a base of \(2\) by rewriting \(10\) as \(2\) multiplied by \(5\): \[2^5 \cdot (2 \cdot 5)^{36}\]
03

Simplify the expression further

Using the power rule, the expression becomes: \[2^5 \cdot 2^{36} \cdot 5^{36}\]
04

Combine the powers of 2

Now we combine the powers of \(2\) together: \[2^{5+36} \cdot 5^{36}\] \[2^{41} \cdot 5^{36}\]
05

Compare the expressions

Now we compare the two expressions, \(2^{125}\) and \(2^{41} \cdot 5^{36}\). Since \(2^{41} \cdot 5^{36}\) has a \(5^{36}\) term, we know that \(2^{41} \cdot 5^{36}\) is larger than \(2^{125}\), because \(5^{36}\) is a large positive number. Therefore, \(32 \cdot 10^{36}\) is larger than \(2^{125}\).

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